WEBVTT
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Language: en
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This is the stochastic processes module 7;
Brownian motion and its properties. Lecture
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1; definition and properties.
In the last six models we started with the
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review of probability. It's one model. Then
the second model we discussed the definition
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of the stochastic process and its properties
and in the third model we have discussed the
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stationary process and all the properties.
Fourth model we have discussed the the discrete-time
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Markov chain and in the fifth model we have
discussed the continuous-time Markov chain.
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In the sixth model we have discussed the [Indiscernible]
[00:01:22] and this is a seventh model that
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is Brownian motion and its properties.
In this lecturer, in this model we are planning
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to discuss the important stochastic process
that is Brownian motion and then later we
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are going to discuss the process derived from
the Brownian motion. Then we are going to
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discuss the stochastic calculus and followed
by that we are going to discuss the stochastic
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differential equation and Ito integrals and
application of the Brownian motion stochastic
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calculus that is in the financial mathematics.
So we are going to discuss the applications
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of a Brownian motions in the financial mathematics
so with that the module seven will be completed.
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And this is the lecture one of - this is the
lecture 1 of module 7 Brownian motion and
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its application. In this lecture we are going
to discuss the the random walk and the definition
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of Brownian motion. Then how one can derive
the Brownian motion using a random walk and
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some important properties of Brownian motions
also will be discussed.
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The
long studied model known as a Brownian motion
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is named after the English botanist Robert
Brown. In 1827 Brown described the unusual
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motion exhibited by a small particle that
is totally immersed in a liquid or gas. It
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is introduced to model the price movements
of stocks and commodities. A formal medical
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description of Brownian motion and its properties
was first given by the great mathematician
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Norbert Wiener beginning in 1918. Therefore
the Brownian motion is also called as Wiener
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process.
Now we start with the random walk because
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using this random walk we are going to derive
the Brownian motion. Consider a trial whose
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outcomes are success with the probability
p or failure with the probability 1 minus
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p. Repeat the trial infinitely many times
that is equivalent of saying tossing fair
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coin infinitely many times. The successive
outcomes are denoted by the sample w that
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consist of w1, w2, w3, where each one is the
outcome in the nth trial. That means w1 could
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be head or tail similarly and w2 could be
head or tail and so on. For example we have
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given H, T, or T, H, T and so on. So this
collection is the - this all the possible
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w's that is going to be the sample space.
Now we are defining the random variable Xj
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it takes the value 1 if the outcome of the
jth trial is head. If the outcome of the jth
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trail is tail then the value is defined for
Xj is minus 1. So this is a real-valued function
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and this will be a random variable since it
takes a value 1 or minus 1 this is a discrete
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random variable and one can find what is a
probability mass function for the random variable
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Xj. So since the trial whose outcomes are
success with the probability p success is
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nothing but the getting head and the failure
is nothing but trial land up with tail. Therefore
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the probability of Xj is equal to 1 that probability
is call it the Wj is equal to head that call
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it as a success therefore this probability
is p and the probability of Xj is equal to
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minus 1 that is 1 minus p and this you can
denote it by q therefore p plus q is equal
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to 1 and denote 1 minus p as quarter ends
as p plus quarter equal to 1. Now we are defining
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the sequence of other random variables that
is started with S naught equal to 0 we are
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defining sum of first k random variables as
a Sk where k is running from 1, 2 and so on.
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Here Xi's are iid random variables. And the
sequence of random variables Sk that is the
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random walk. With the S naught is equal to
0 and the Sk's are nothing but the first k
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Xi random variables.
You can see the sample path of the random
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bar whose w1 is a tail therefore takes a value
X1 is minus 1. Again if the suppose w2 is
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t then X2 also takes the value minus 1. Suppose
w3 is also T then X3 also takes minus 1. Therefore
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Sk will be initially it is 0 then S1 will
be X1 that is minus 1. S2 will be X1 plus
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X2 that is minus 1 plus minus 1 that is minus
2. So S2 is minus 2. S3 will be S2 plus X3
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that is a again adding minus 1 so S3 will
be minus 1. Like that it can take the different
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values. So here this is a one sample path
with the w1 is equal to T and w2 is equal
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to T and w3 is equal to T and so on with the
probability P is equal to 0.45. This is the
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probability of success or probability of getting
head when you toss a coin.
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So we are going to conclude later as k tends
to infinity using central limit theorem one
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can conclude this will be a Brownian motion.
For that you should understand how the Sk's
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are are created wher Sk's are the sample path,
where Sk's are the random walk.
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Now we are going to see the properties of
random walk. If you choose non-negative integers
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k naught, k1 and so on not k1 and so on then
if you find that difference the difference
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is nothing but a sum of Xi's in this range.
Since the Xi's are iid random variables if
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you take a non-overlapping intervals or the
increments of Si's then that will be mutual
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[Indiscernible] [00:10:06] because each these
increments will be nothing but the sum of
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a few Xi's and we know that each Xi's are
mutually independent iid random variables
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therefore non-overlapping increments will
be mutually independent random variables.
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Hence Sn has the property called the independent
increment, the the increments of independence.
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Similarly for 0 less than or equal to i less
than or equal to j Si minus Sj is identically
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distributed with Sj plus H minus Si plus H
for H belonging to natural numbers. Hence
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the stochastic process Sn has stationary increment
property. That means if you find out the n
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dimensional random variable and shifted by
H find out the another n dimensional random
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variable if though the joint distributions
are same for both the n dimension random variable
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without shifting and with shifting then that
stochastic process is called stationary but
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here the stochastic process is not a stationary
the increments are stationary means they have
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increments and you shifted the increment by
some interval H then the distributions are
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going to be identical. That's what it shows
for one less than or equal to i less than
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j less than or equal to k less than l the
difference the distributions are going to
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be same as long as their length is same. So
it is in the increments are time invariants
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not the actual stochastic process. Therefore
this stochastic process has the stationary
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increment also. Therefore, the random walk
has increments are stationary as well as invariants.
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Also one can find a mean and variance of increments.
The increments are nothing but the difference
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of those random variables and since each random
variable are discrete type random variable
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with the probability mass function that is
discussed in the previous slide so we can
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find out the mean and variance of those random
variables therefore we can find out the mean
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and variance of increments also.
Now we are going to derive the Brownian motion
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using random walk. Consider a particle performs
a random walk such that in a small interval
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of time of duration delta t the displacement
of the particle to the right or to the left
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is also a small magnitude Delta x whenever
a particle performs a random walk in a very
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small interval of time delta t the displacement
of a particle to the right or to the left
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that magnitude is Delta x. Now we are defining
a random variable S of t denotes the total
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displacement of the particle in time t. let
Xj denote the length the jth taken by the
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particle in a small interval of time delta
t with the probability mass function. So the
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probability of Xj takes the displacement of
the particle to the right side that is Delta
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x with the probability p the left side that
is the Xj takes the value minus Delta x that
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is 1 minus p that is nothing but a q where
p plus q is equal to 1 where p is independent
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of X as well as time. It is very important.
The probability of the displacement to the
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right or to the left that probability whether
p or 1 minus p which is independent of X as
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well as time. Now the partition of the interval
of length t into n equal subintervals of Delta
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x. Then the n times delta t becomes t and
the total displacement St is the sum of n
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iid random variables Xt. The way we partition
the interval 0 to t into n equal parts therefore
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the S of t the total displacement is nothing
but the sum of n iid random variables Xj's
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where n is nothing but n of t because you
are partitioning the interval n sorry you
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are partitioning the time interval 0 to t
the length of t into n parts therefore n is
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nothing but n of [Indiscernible] [00:15:41]
that is nothing but t divided by Delta t.
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So you know the mean and variance. Therefore
you can find out the mean and variance of
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S of t also. because St is a sum of n iid
random variables Xj. Expectation is a linear
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operator therefore n since it is iid random
variable n times expectation of any one random
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variable whereas variance since the random
variables are independent then the variance
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of S of t is nothing but the variance of sum
of random variables. So you can take it out
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and you can do the simplification.